A function f(x) is defined as follows : `f(x)={(x sin ""1/x ", " x ne 0),(0 ", " x=0):}
Discuss its continuity at x=0

Explanation

At x=0
f(0)=0
R.H.L `=underset(xrarr0^+)limf(x)=underset(hrarr0)limf(0+h)`
`=underset(hrarr0)limhsin"" (1)/(h)`
`=0xx`(a definite value )
=0
`(therefore) "sin" (1)/(h) ` always lies between -1 and 1)
gt L.H.L `=underset(xrarr0^-)lim f(x)=underset(hrarr0)limf(0-h)`
` =underset(hrarr0)lim(-h)sin (1/-h)=underset(hrarr0)(lim) h sin ""1/h`
`=0xx` (a definite value )=0
`therefore` R.H.L = f(0)=L.H.L
`therefore` f(x) is continuous at x=0.
Prove that(x) is continous at x=2.